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How does mainstream climate change science arrive at CO₂ = catastrophic warming?

Nobody is reading thousand-page IPCC reports. People just trust headlines, actors, etc.

I'm not that trusting For myself I had to create this primer to connect it all together. [0/n]
Any warm object radiates energy at wavelengths depending on its temperature. The distribution of wavelengths is described by Planck's Law. [1/n]
Planck's Law predicts that the Sun, having a surface temperature of 5,800 degrees, will radiate mostly at visible light wavelengths. [2/n]
The sun provides an enormous amount of energy. Earth receives 1,367 W/m² from the sun. [3/n]

Image credit: windows2universe.org/earth/climate/…
Earth reflects and radiates a total of 342 W/m². This is exactly ¼ of the solar flux, because the sun illuminates a disk (πr²) but the Earth radiates from a spherical surface (4πr²). This is "radiative equilibrium" and is expected in the long-term. [4/n]
According to Planck's Law, the Earth, being much cooler than the sun, will radiate mostly longwave infrared. [5/n]
The area under the curve described by Planck's Law is the radiant flux (W/m²) which may be computed using a simpler equation known at the Stefan-Boltzmann Law. [6/n]
If the Earth had no atmosphere, it would radiate infrared to space across a smooth range of wavelengths. Instead, Earth has an atmosphere that absorbs & re-radiates some outgoing infrared causing the final emission to space to have a noisy, jagged form. [7/n]
IR-active gasses suppress emissions in portions of the spectrum, so others shift upward to compensate: surface warming. The surface now emits ~382 W/m², though only 342 W/m² escapes to space. The difference between these two numbers (~40 W/m²) is the greenhouse effect (GHE). 8/n
CO₂ concentration rising from 369 to 670 ppm is predicted to increase the GHE by +3.2 W/m². [9/n]
+3.2 W/m² from +300 ppm CO₂ causes additional surface and lower atmosphere warming. Warmer air can hold more water vapor, a greenhouse gas. The IPCC asserts that for every 1 W/m² CO₂ forcing, water vapor amplifies this "by 2-3x." CO₂'s +3.2 W/m² becomes +6.4-9.6 W/m². [10/n]
The climate sensitivity parameter describes how many degrees the Earth warms up given additional forcing from doubled CO₂. This is a difficult number to know with certainty; over time researchers have produced a range of estimates typically falling between 1.5 and 4.5C. [11/n]
+3.2 W/m² & no H₂O vapor feedback & lower (but empirically-supported) climate sensitivity parameter of 1.6 predicts a habitable planet at 670 ppm CO₂. But a 3x H₂O feedback plus a stronger (~4) climate sensitivity parameter predicts catastrophic warming (+9.5C). [12/n]
Those who anticipate catastrophic warming in their lifetime are implicitly accepting & expecting a high water vapor feedback parameter and a strong climate sensitivity parameter. [13/n, n=13]
If you find anything to be factually incorrect, please let me know (+source). My goal is complete accuracy.

The evergreen version I continually update (with all source citations that didn't fit in tweets) is here: chrisrentsch.com/home/GW
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